Actual electrical currents I suppose are not always along strict submanifolds -- they can be along all kinds of shapes, like in a lightning strike. So it's probably a better match with reality to consider their data as carried in the distributional sense of functionals on forms. I imagine your intuition is correct but I'm not so certain about the specific history. There is likely people on this forum that know the history well but if not I'd suggest asking someone like Cartier or Berger.
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Ryan BudneyApr 10 '13 at 19:04

Thank you for turning me on to Lützen's book! I had turned to de Rham's book before posting this question (rather an English translation thereof), this book gives a sense of the big picture.
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D. KelleherApr 11 '13 at 3:16

The classical electric current density can be modelled as a 2-form
$$J=J_{ij}\wedge dx^{ij}$$
which is assumed to be locally integrable over a 3-manifold (3-dimensional domain) $X$. By integrating $J$ over a 2-submanifold (a surface) $S\subset X$, one gets the electric current through S, i.e.
$$I(S)=\int_{S}J,$$
that is a simple example of a 2-current in the sense of de Rham. Extending this notion to the $n$-dimensional case, one can naturally model a density on the $n$-manifold as a twisted $(n − 1)$-form $J\in\Omega^{n-1}(X,L)$, and to treat the corresponding integral $I(S)$ as a generalized electric current through the (n-1)-dimensional submanifold $S\subset X$.

Whether this was a real motivation behind the notion of de Rham's currents, I don't know.

I am not sure if this helps, but elaborating on Andrey's answer: now if you recall that occasionally one wants to work with electric currents localised on submanifolds of $\mathbb{R}^3$, what a physicist would call a "wire", or a "conducting plane", de Rham currents provide a natural framework for this. This is much like using distributions to describe point charges.

For example, a current corresponding to a loop wire carrying electric current $I$ represented by an oriented curve $\gamma$ is the 1-current $J$ given by $J(\phi)=I\int_\gamma\phi$. This can be though of as a "generalized" 2-form $J$, which we now can try to integrate over 2-manifold $S$ to obtain the current through this surface, but I am not sure that it makes rigourous sence for arbitrary 1-currents (for this particular one it does, at least for finite number of transversal intersections of $\gamma$ and $S$). We can also take $J(E)=RI^2$, where R is the resistance, and $E$ is the electric field 1-form to express the Ohm's law, et cetera.